We construct the global bounded solutions
and the attractors of a parabolic-parabolic chemotaxis-growth system
in two- and three-dimensional smooth bounded domains.
We derive new $L_p$ and $H^2$ uniform estimates for these solutions.
We then construct the absorbing sets and the global attractors
for the dynamical systems generated by the solutions.
We also show the existence of exponential attractors
by applying the existence theorem of Eden-Foias-Nicolaenko-Temam.